Stochastic Integro-differential Equations Research Articles

Existence for a Second-Order Impulsive Neutral Stochastic Integrodifferential Equations with Nonlocal Conditions and Infinite Delay

The current paper is concerned with the existence of mild solutions for a class of second-order impulsive neutral stochastic integrodifferential equations with nonlocal conditions and infinite delays in a Hilbert space. A sufficient condition for the existence results is obtained by using the Krasnoselskii-Schaefer-type fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. Finally, an application to the stochastic nonlinear wave equation with infinite delay is given.

Asymptotic stability of impulsive stochastic partial integrodifferential equations with delays

In this paper, we study the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear impulsive stochastic partial functional integrodifferential equations with delays. We suppose that the linear part possesses a resolvent operator in the sense given in Grimmer [R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Am. Math. Soc. 273(1) (1982), 333–349] and the nonlinear terms are assumed to be Lipschitz continuous. A fixed point approach is employed for achieving the required result. An example is provided to illustrate the results of this work.

Nonlocal Cauchy problem for fractional stochastic evolution equations in Hilbert spaces

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then \(n\)-dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of \(n\) action variables and \(n\)- independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.

Existence of Square-Mean Almost Automorphic Solutions to Stochastic Functional Integrodifferential Equations in Hilbert Spaces

The existence and uniqueness of square-mean almost automorphic mild solution to a stochastic functional integrodifferential equation is studied. Under some appropriate assumptions, the existence and uniqueness of square-mean almost automorphic mild solution is obtained by Banach’s fixed point theorem. Particularly, based on Schauder’s fixed point theorem, the existence of square-mean almost automorphic mild solution is obtained by using the condition which is weaker than Lipschitz conditions. Finally, an example illustrating our main result is given.

Robust exponential stability of mild solutions to impulsive stochastic neutral integro-differential equations with delay

Robust exponential stability of mild solutions to impulsive stochastic neutral integro-differential equations with delay

Block pulse approximation of fractional stochastic integro-differential equation

The aim of the presented paper is to apply block pulse operational matrices for solving fractional stochastic integro differential equations. By using this approach, the fractional stochastic integro-differential equation reduces to a linear system of algebraic equations which can be solved by iterative method. Accuracy and efficiency of the method are shown with an example.

Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-partially integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. For the case with r(1<r<n) independent first integrals which are in involution, an r-dimensional averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for the transition probability density of r independent first integrals is derived from the stochastic integro-differential equations (SIDEs) of the original quasi-partially integrable and non-resonant Hamiltonian systems by using the stochastic jump-diffusion chain rule and the stochastic averaging theorem. An example is given to illustrate the applications of the proposed stochastic averaging method, and a combination of the finite difference method and the successive over-relaxation method is used to solve the reduced GFPK equation to obtain the stationary probability density of the system. The results are well verified by a Monte Carlo simulation.

Exponential stability for some stochastic neutral partial functional integrodifferential equations with delays and Poisson jumps

In this work, we study the existence, uniqueness and exponential stability in mean square of mild solutions for some stochastic neutral partial functional integrodifferential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square of mild solutions are derived by means of the Banach fixed point principle. We suppose that the linear part has a resolvent operator in the sense given in Grimmer (Trans. Am. Math. Soc., 273(1):333–349, 1982). An example is provided to illustrate the results of this work.

Parabolic Littlewood–Paley inequality for [formula omitted]-type operators and applications to stochastic integro-differential equations

In this paper we prove a parabolic version of the Littlewood–Paley inequality (1.4) for the operators of the type ϕ(−Δ), where ϕ is a Bernstein function. As an application, we construct an Lp-theory for the stochastic integro-differential equations of the type du=(−ϕ(−Δ)u+f)dt+gdWt.

Semilinear Neutral Fractional Stochastic Integro-Differential Equations with Nonlocal Conditions

In this paper, we prove the existence of mild solutions for a class of semilinear neutral fractional stochastic integro-differential equations with nonlocal conditions. Sufficient conditions for the existence are derived with the help of the Sadovskii fixed point theorem. In the end, an example is given to show the application of our results.

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

Square-mean almost periodic solutions to some stochastic evolution equations

This paper concerns the square-mean almost periodic mild solutions to a class of abstract nonautonomous functional integro-differential stochastic evolution equations in a real separable Hilbert space. By using the so-called “Acquistapace-Terreni” conditions and the Banach fixed point theorem, we establish the existence, uniqueness and the asymptotical stability of square-mean almost periodic solutions to such nonautonomous stochastic differential equations. As an application, almost periodic solution to a concrete nonautonomous stochastic integro-differential equation is considered to illustrate the applicability of our abstract results.

Numerical solution of a stochastic population growth model in a closed system

In this paper, we introduce a stochastic population model in a closed system. This model is a nonlinear stochastic integro-differential equation. At first, we solve this problem via the stochastic θ-method. Then we solve it by using the Bernstein polynomials and collocation method. This method reduces integro-differential equation to a system of nonlinear algebraic equations. The results demonstrate applicability and accuracy of this method.

A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion

This paper considers the solution of the stochastic integro-differential equation of Milne problem with random operator. The Pomraning–Eddington method is implemented to get a closed form solution deterministically. Relying on the spectral properties of the covariance function, the Karhunen–Loeve (K–L) expansion is used to represent the input stochastic process in the deterministic solution. This leads to an explicit expression for the solution process as a multivariate functional of a set of uncorrelated random variables. By using different distributions for these variables, the work is realized through computing the mean and the variance of the solution. The numerical results are found in agreement with those obtained in the literature.

Almost Automorphic Solutions to Nonautonomous Stochastic Functional Integrodifferential Equations

This paper concerns the square-mean almost automorphic solutions to a class of abstract semilinear nonautonomous functional integrodifferential stochastic evolution equations in real separable Hilbert spaces. Using the so-called “Acquistapace-Terreni” conditions and Banach contraction principle, the existence, uniqueness, and asymptotical stability results of square-mean almost automorphic mild solutions to such stochastic equations are established. As an application, square-mean almost automorphic solution to a concrete nonautonomous integro-differential stochastic evolution equation is analyzed to illustrate our abstract results.

Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

Advanced Theoretical and Applied Studies of Fractional Differential Equations

The fractional calculus (derivatives and integrals of arbitrary order) has developed into an effectivemodelingmethodology for a variety of real-world problems. Based on the wide applications in engineering and sciences such as physics, mechanics, chemistry, and biology, research on fractional ordinary and partial differential equations and other related topics is active and extensive around the world. In the past few years, the increase in the popularity of the subject can be witnessed by hundreds of research papers published in prestigious journals, several monographs, and international conferences and workshops. The special issue, focused on the importance of fractional operators and their applications, received 65 papers. Only 33 highest quality papers were accepted for publication after the peer review process. The research papers in this special issue cover various topics like numerical methods for fractional partial differential equations, practical sketching rules for fractional order systems, nonlocal boundary value problems of fractional qdifference equations, fractional steepest descent approach for modeling texture imaging, fractional order chaotic systems, fractional order circuits, approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems, fractional abstract Cauchy problems, positive solutions for a new coupled system of multiterm singular fractional integrodifferential equations, and asymptotic behavior of solutions to abstract stochastic fractional partial integrodifferential equations.

Stability and Boundedness of Stochastic Volterra Integrodifferential Equations with Infinite Delay

We make the first attempt to discuss stability and boundedness of solutions to stochastic Volterra integrodifferential equations with infinite delay (IDSVIDEs). By the Lyapunov-Krasovskii functional approach, we get kinds of sufficient criteria for stability and boundedness of solutions to IDSVIDEs. The main innovation here is that stochastic systems with infinite delay can retain stability and boundedness of corresponding deterministic systems under some conditions.

Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for approximately predicting the responses of multi-degree-of-freedom (MDOF) quasi-non-integrable Hamiltonian systems (non-integrable Hamiltonian systems with lightly linear and (or) nonlinear dampings) to weakly external and (or) parametric excitations of Gaussian and Poisson white noises is proposed. By means of the stochastic integro-differential equations (SIDE) and stochastic jump-diffusion chain rule, a one-dimensional averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for the transition probability density of the Hamiltonian is derived to obtain the probability density and statistics of the stationary responses through solving the GFPK equation using perturbation technique. An example is given to illustrate the application of the proposed method. It is shown that theoretical results agree well with those from Monte Carlo simulation.